Application of the Krylov-Bogoliubov-Mitropolski Technique for a Rotating Heavy Solid under the Influence of a Gyrostatic Moment

Advances in Mechanical Engineering

2017

Self Cite

This work shed light on the motion of a symmetric rigid body (gyro) about one of its principal axes in the presence of a Newtonian force field besides a gyro moment in which its second component equals null. It is assumed that the body center of mass is shifted slightly relative to the dynamic symmetry axis. The governing equations of motion are investigated taking into account some initial conditions. The desired solutions of these equations are achieved in framework of the small parameter method. The periodic solutions for the case of irrational frequencies are investigated. Euler's angles have been used to interpret the motion at any time. The geometrical representations of the obtained solutions and the phase plane schemas of these solutions are announced during several plots. Discussion of the results is presented to reinforce the importance of the considered gyro moment and the Newtonian force field. The significance of this problem is due to the framework of its several applications in different industries such as airplanes, submarines, compasses, spaceships, and guided missiles.

Section: Description Of the Problem And The Proposed Methodsmentioning

confidence: 99%

On the dynamical motion of a gyro in the presence of external forces

Advances in Mechanical Engineering

2017

Self Cite

“…In view of the relation between the derivatives of F from second order, we can rewrite equation (10) in the form…”

Section: Proofmentioning

confidence: 99%

“…Moreover, the Krylov–Bogoliubov–Mitropolski method has been applied [10] to attain solutions of the equations of motion for the motion of a rigid body in a uniform field besides the action of the gyrostatic moment on the third rotating axis. This problem was generalized [11] using the same method to obtain the desired analytical solutions, which have been verified through comparison with corresponding numerical solutions.…”

Section: Introductionmentioning

confidence: 99%

Substantial condition for the fourth first integral of the rigid body problem

Mathematics and Mechanics of Solids

2017

The aim of this article is to study the possibility of obtaining the fourth integral for the motion of a rigid body about a fixed point in the presence of a gyrostatic moment vector. This problem is governed by a system consisting of six nonlinear differential equations from first order, as well as three first integrals. A most important condition for a function F, depending on all the body variables, to be that integral is presented. This work can be considered a mainstreaming of previous works. The importance of this work lies in several applications of the rigid body problem and gyroscopic motion in different areas, such as physics, engineering and industrial applications, for example, in aircraft specially designed to use the auto-pilot function, calculating aircraft turns about various axes of operation (pitch, yaw and roll), and maintaining aircraft orientation.

“…On the other hand, this motion is studied in the work by Amer [8] when the body is subjected to the influence of a gravitational field and a constant gyro moment. Moreover, this problem is solved in the work by Amer et al [9] using the Krylov -Bogoliubov -Mitropolski (KBM) technique to obtain the approximate periodic solution. In the work by Cavas and Vigueras [10], the authors investigated the analytical solutions for a gyrostat similar to Lagrange’s case under the action of a Newtonian field.…”

Section: Introductionmentioning

confidence: 99%

“…The dynamical motion of a rigid body is considered as a core subject of the field of theoretical mechanics and it has great interest from a scientific and practical point of view. The solution of the rigid body problem has been studied in a great number of scientific works see monographs [1][2][3][4][5][6][7][8][9][10][11]. As known, this problem has three first integrals related with energy, area and geometric integrals [1].…”

Section: Introductionmentioning

confidence: 99%

On the motion of a gyro in the presence of a Newtonian force field and applied moments

Mathematics and Mechanics of Solids

Abady

2017

This work focuses on the motion of a dynamical model that consists of a symmetric rigid body (gyro) that rotates about a fixed point similar to Lagrange’s gyroscope. This body is acted upon by external forces represented by a Newtonian force field, gyro torques about the principal axes of inertia of the gyro and perturbing moments acting on the same axes. Assuming that, the gyro initially has a high angular velocity about the dynamic axis of symmetry. The averaging technique is used to obtain a more appropriate averaging system for the governing system of equations of motion in terms of a small parameter. Therefore, the analytical solutions of this system for two applications, depending on different forms of perturbing moments, are presented. These solutions are represented graphically to clarify the effectiveness of the different parameters of the body on the motion.